Classifying Functions — One-One, Many-One, Onto and Bijective

Once a diagram is confirmed to represent a function, it can be classified further by how its arrows are arranged.

A function is one-one (or injective) if no two different elements of the domain ever share the same arrow-target — every element of the range receives exactly one arrow. It is many-one if at least one element of the co-domain receives arrows from two or more different domain elements — the direct opposite of one-one.

A function is onto (or surjective) if its range equals its entire co-domain — every element of $B$ receives at least one arrow, with none left out. If some element of the co-domain is left without any arrow at all, the function is called into instead.

A function that is both one-one and onto is called bijective — every element of $A$ pairs with a distinct element of $B$, and every element of $B$ is used exactly once. A bijective function pairs the two sets up perfectly, with no element left over on either side and no element of $B$ shared between two elements of $A$; in particular, a bijective function between two finite sets requires $n(A) = n(B)$.

Worked Example 11 — One-One and Onto → Bijective:

Exercise 7, part (i) — Many-One and Onto:

Exercise 7, part (ii) — One-One and Into: